Shooting method

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In numerical analysis, the shooting method is a method for solving a boundary value problem by reducing it to the solution of an initial value problem. The following exposition may be clarified by this illustration of the shooting method.

For a boundary value problem of a second-order ordinary differential equation, the method is stated as follows. Let

<math> y(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y(t_1) = y_1 </math>

be the boundary value problem. Let y(t₁; a) denote the solution of the initial value problem

<math> y(t) = f(t, y(t), y'(t)), \quad y(t_0) = y_0, \quad y'(t_0) = a </math>

Define the function F(a) as the difference between y(t₁; a) and the specified boundary value y₁.

<math> F(a) = y(t_1; a) - y_1 \,</math>

If the boundary value problem has a solution, then F has a root, and that root is just the value of y'(t₀) which yields a solution y(t) of the boundary value problem.

The usual methods for finding roots may be employed here, such as the bisection method or Newton's method.

Linear shooting method

The boundary value problem is linear if f has the form

<math> f(t, y(t), y'(t))=p(t)y'(t)+q(t)y(t)+r(t). \, </math>

In this case, the solution to the boundary value problem is usually given by:

<math>y(t) = y_{(1)}(t)+\frac{y_1-y_{(1)}(t_1)}{y_{(2)}(t_1)}y_{(2)}(t)</math>

where <math>y_{(1)}(t)</math> is the solution to the initial value problem

<math>y(t) = f(t, y(t), y'(t)),\quad y(t_0) = y_0, \quad y'(t_0) = 0, </math>

and <math>y_{(2)}(t)</math> is the solution to the initial value problem:

<math>y(t) = p(t)y'(t)+q(t)y(t),\quad y(t_0) = 0, \quad y'(t_0) = 1. </math>

See the proof for the precise condition under which this result holds.

References

• Josef Stoer and Roland Bulirsch. Introduction to Numerical Analysis. New York: Springer-Verlag, 1980. (See Section 7.3.)